Introduction to Combinatorial Methods in Geometry

This book offers an introduction to some combinatorial (also, set-theoretical) approaches and methods in geometry of the Euclidean space R*m*. The topics discussed in the manuscript are due to the field of combinatorial and convex geometry.

Autorentext

Alexander Kharazishvili is Head of the Department of Discrete Mathematics at I. Vekua Institute of Applied Mathematics.

Inhalt

1. The index of an isometric embedding. 2. Maximal ot-subsets of the Euclidean plane. 3. The cardinalities of at-sets in a real Hilbert space. 4. Isosceles triangles and it-sets in Euclidean space. 5. Some geometric consequences of Ramsey's combinatorial theorem. 6. Convexly independent subsets of infinite sets of points. 7. Homogeneous coverings of the Euclidean plane. 8. Three-colorings of the Euclidean plane and associated triangles of a prescribed type. 9. Chromatic numbers of graphs associated with point systems in Euclidean space. 10. The SzemerediTrotter theorem. 11. Minkowski's theorem, number theory, and nonmeasurable sets. 12. Tarski's plank problem. 13. Borsuk's conjecture. 14. Piecewise affine approximations of continuous functions of several variables and CaratheodoryGale polyhedral. 15. Dissecting a square into triangles of equal areas. 16. Geometric realizations of finite and infinite families of sets. 17. A geometric form of the Axiom of Choice.